Logical Analysis:
Propositional calculus is a method of applying logic to arguments to test their validity. Arguments are not always presented in a nice easy format, like those examples I have given in my Logic writeup, such as:
1) All men are mortal
2) Socrates is a man
3) Therefore Socrates is mortal
Often they will be contained in
prose and it may not always be obvious or easy to tell if an argument we encounter is valid or not. As I said, the
logical analysis of propositional calculus provides us with a system that we can use to test an argument's validity.
In order for our system to work, we need some sort of standard format for presenting arguments. We need to "formalise" them, which we do by breaking up arguments into their constituent parts, or sentences, which we represent by a letter, like in algebra. We then present these as "truth functors".
Truth Functors:
Truth functors assign notions of truth to sentences so that we can test validity. There are five basic truth functors, each defined by a truthtable. Truthtables are simply a way of showing when an expression is true and false, given the truthvalue of its constituent parts. It is clear what they are from simply looking at them  an extensive explanation of what they are is not necessary here.
The basic truth functors are "and", "or", "if, then", "if and only if", and "not". Each have their own symbol.
AND:
"And" is represented by the symbol " ^ ". So "A and B" would be written "A ^ B". The truthtable for AND is:
A  B  A ^ B
T  T  T
T  F  F
F  T  F
F  F  F
A ^ B requires both A to be true and B to be true. Obviously, if one or both of A or B are false, then one cannot have both of them true, which is what the AND functor requires. AND is used in cases where all the individual parts of the sentence need to be true for the whole sentence to be true. Some examples of sentences that would be written "A ^ B" are:
"A and B" = e.g. "The soup has salt and pepper in it."
"A but B" = e.g. "The soup has salt in it, but it has pepper in too."
"A as well as B" = e.g. "The soup has salt in as well as pepper."
"Although A, B" = e.g. "Although the soup has salt in it, it has pepper in."
There can be some uses of the word "and" in
English that should not be formalised using " ^ ". Some examples are:
"London and Edinburgh are 300 miles apart."
Which would come out as; "London is 300 miles apart ^ Edinburgh is 300 miles apart"  which is gibberish!
Similar instances include:
"The film is in black and white" (which would become; "The film is in black ^ The film is in white")
"Twenty people were rounded up and shot" (which would become; "Twenty people were rounded up ^ twenty people were shot")
Both of these mean different things that the original sentence.
OR:
"Or" is represented by the symbol " v ". "A or B" would be written " A v B". The OR truthtable is:
A  B  A v B
T  T  T
T  F  T
F  T  T
F  F  F
There is some debate here as to how we interpret OR. We can either read it "weakly" or "strongly". The weak reading (which is what we use) says that if both A and B are true then "A or B" is true. However, the strong reading would say that since they are both true, then we cannot have either A or B  we have to have both. The strong reading says that "A or B" is false. This is an interesting debate, but for the purposes of logical analysis, the weak reading is required. This is the only part of the truthtable that is not really obvious.
There are not many types of sentences in English which apply to "OR". They will often simply contain "or" within them, so will be easy to spot. Examples which would be written "A v B" include:
"A or B" = e.g. "The soup is tomato or minestrone."
"A unless B" = e.g. "The soup is tomato, unless it is minestrone."
Very occasionally, "or" can be more analogous to "and" when used in sentences like:
"The scarf can be worn by boys or girls."
So one must
be careful.
IF, THEN:
"If, then" is represented by the symbol "→". So, "If A, then B" would be written "A → B". The truthtable for "IF, THEN" is:
A  B  A → B
T  T  T
T  F  F
F  T  T
F  F  T
We can see how this truthtable is derived from thinking about possible worlds and arguments. It is not logically possible for a set of premises to be true, and their conclusion false. Arguments are themselves a form of an "IF, THEN" statement. So with this in mind, we can see that the only way that "If A, then B" can be false is if A is true, and B is false, and hence the truthtable is as above. One might ask, "well, if both A and B are false, then how can "if A, then B" be true?". Well, this is a good question, but in our logical system, it is a true statement as there is not really any contradiction here. One could also ask whether "if A, then B" is true when A is false and B is true. But since anything follows a contradiction, the statement is true.
"If, then" is not always easy to apply to actual sentences in English. However, some examples which would be written "A → B" are:
"If A, then B" = e.g. "If the paper turns red, then the solution is acidic"
"A only if B" = e.g. "The paper will turn red only if the solution is acidic"
"B provided that A" = e.g. "The solution is acidic provided that the paper turns red"
"Assuming that A, B" = e.g. "Assuming that the paper turns red, the solution is acidic"
Examples that should not be interpreted as an "→" statement are things like:
"The soup was nice, if a little salty."
Here, "if" is not acting in any conditional sense. It is used in the same way as "and";
"The soup was nice, and the soup was salty."
Other examples of sentences which use "if", but should not be interpreted using "IF, THEN" are:
"If you want some soup, I have some in the pantry."
"I wouldn't eat that soup if you paid me!"
IF AND ONLY IF:
"If and only if" is represented by the symbol "↔", so "A if and only if B" would be formalised as "A ↔ B". IF AND ONLY IF is, I suppose, a more specific sort of conditional than IF, THEN. "If and only if" is really a combination of two "if" statements. So, "A ↔ B", is equivalent to "A → B" and "B → A", or more formally; "[A → B] ^ [B → A]" (the square brackets are what we use to separate the parts of the logical formulae). Here we treat the "if, then" formalisations as two clauses separated by the "and" truth functor. The derivation of the IF AND ONLY IF truthtable provides a good example as to how we combine truthtables.
Firstly, we start with the two truthtables for [A → B] and for [B → A]:
A  B  A → B
T  T  T
T  F  F
F  T  T
F  F  T
A  B  B → A
T  T  T
T  F  T
F  T  F
F  F  T
Then, we combine these with the aid of the AND truthtable. To do this, we look at the truth values of the two formalisations above and match them up with the AND truthtable, to see what the final truth value of the whole formula is.
A → B  B → A  [A → B] ^ [ B → A]
T  T  T
F  T  F
T  F  F
T  T  T
So:
A  B  A ↔ B
T  T  T
T  F  F
F  T  F
F  F  T
We can see from this derivation why the truthtable for IF AND ONLY IF is the way it is. Some English phrases that capture the sense of this truth functor, and would be formalised "A ↔ B" are:
"A if and only if B" = e.g. "A number is even if and only if it is divisible by two"
"A precisely if B" = e.g. "A number is even precisely if it is divisible by two"
"A just if B" = e.g. "A number is even just if it is divisible by two"
In the
USA, but not really in
Britain, "
just in case" can be used in the same sense as "if and only if". It is important to note that "only if" is not the same as "if and only if". "Only if" is actually equivalent to "if" on its own  the "only" is there for
emphasis in speech.
NOT:
"Not" is symbolised by "¬" and precedes that which is to be negated. So "not A" would be formalised as "¬A". "¬A" would be pronounced as "it is not true that A". The NOT functor therefore has the effect of reversing the truthvalue of that which it is associated with. The truthtable is simply:
A  ¬A
T  F
F  T
We say that "A" is "the negation of sentence A". We can find very many examples in English:
"I am not French", gives; "¬ I am French"
"I didn't say anything", gives; " ¬ I said something"
"He never went to London", gives; "¬ He went to London"
"I hardly think he is right", gives; "¬ I think he is right"
"It isn't as if he needs the money", gives; "¬ He needs the money"
Each of the other truth functors has a negation, the effect of which is to reverse their truthtables:
A  B  ¬[A ^ B]
T  T  F
T  F  T
F  T  T
F  F  T
A  B  ¬[A v B]
T  T  F
T  F  F
F  T  F
F  F  T
A  B  ¬[A → B]
T  T  F
T  F  T
F  T  F
F  F  F
A  B  ¬[A ↔ B]
T  T  F
T  F  T
F  T  T
F  F  F
A  ¬¬A
T  T
F  F
The formalised logical sentences produced using truth functors are known as propositions. We can use combinations of truth functors to produce strings of propositions as long as needed. Longer and more complex sentences in English will need to be represented by several truth funtors. The truth values of these can be determined using combinations of truthtables, as was done above to resolve the IF AND ONLY IF functor.
Propositional Tableaux:
Truthtables are not a powerful enough tool to analyse the consistency of the sets of statements and beliefs we find in more commonly used English sentences. For this we use a system of analysis known as propositional tableaux, which is part of the propositional calculus used to analyse logical propositions.
Propositional tableaux take the form of a series of branches that progressively split a set of statements into its constituent parts. By doing so we can easily discover if any parts are contradictory.
The input to the tableau takes the form of a series of sentences formalised into the propositional logic. Thus, the rules dictating the formation of the branches of propositional tableaux are derived directly from the truth functors and their truth tables. The rules are derived by looking at the cases where the functor can be true and representing each of these cases with a branch.
AND:
[A ^ B]

A
B
NOT AND:
¬[A ^ B]
________
 
¬A ¬B
OR:
[A v B]
________
 
A B
NOT OR:
¬[A v B]

¬A
¬B
IF, THEN:
[A → B]
________
 
¬A B
NOT IF, THEN:
¬[A → B]

A
¬B
IF AND ONLY IF:
[A ↔ B]
________
 
A ¬A
B ¬B
NOT IF AND ONLY IF:
¬[A ↔ B]
________
 
A ¬A
¬B B
NOT NOT:
¬¬A

A
The best way to illustrate how we use tableaux is to go through an example. Firstly, we have a set of statements:
1) John likes tea if he is English.
2) John is English unless his father is Scottish.
3) John's father is English, but John does not like tea.
We need to convert these sentences using the truth functors of the propositional calculus as the first stage in testing their consistency:
1) [John likes tea → John is English]
2) [John is English v John's father is Scottish]
3) [John's father is English ^ ¬John likes tea]
To make life easier, we symbolise the constituent sentences. We call this an "
interpretation". The
letters are known as "sentence letters":
A: John is English
B: John likes tea
C: John's father is English
D: John's father is Scottish
So our logical propositions are now:
1) [A → B]
2) [A v D]
3) [C ^ ¬B]
The above are called "formulae". We place them in square brackets to group together all the parts which are constituent in the same phrase. It would be important to note at this point that "¬" placed in front of brackets
negates everything inside, whereas if it is placed inside the brackets, it only negates the letter it is associated with. The term for this is the "
scope" of "¬".
We can now construct the tableau. We list the original formulae and make branches off from them. Tableaux can be thought of as an upsidedown tree, with the original formulae as the trunk, with branches expanding downwards. The "truck" of formulae actually form part of the branches though.
To proceed, we continue adding branches on the end of each branch until we can add no more. When we reach a contradiction in a branch, we close that branch by putting a line under it and adding no more branches following it. We continue until we have applied all our propositions, or all the branches of the tableau are closed.
A contradiction occurs only if we find that a branch contains some proposition "X" and also the proposition "¬X"  these cannot both be true.
Armed with our tableau rules, we can now construct our tableau:
First the "trunk":
[A → B]
[A v D]
[C ^ ¬B]
From here we could apply any of the "IF", "OR", or "AND" rules. It doesn't matter which order the rules are applied. We shall start by applying the "OR" rule to the second formula:
[A → B]
√ [A v D]
[C ^ ¬B]
________
 
A D
When a rule is applied to a formula, that formula cannot be used again. To help keep track, we usually place a tick next to it. Each time we apply a rule we must check to see if there are any contradictions to see if the tableau can be closed. We have none yet.
Now we shall apply the "AND" rule to the end of each branch:
[A → B]
√[A v D]
√[C ^ ¬B]

________
 
A D
C C
¬B ¬B
Again, we have no contradictions. Now to apply our last rule, the "IF" rule:
√ [A → B]
√ [A v D]
√ [C ^ ¬B]

________
 
A D
C C
¬B ¬B
______ ______
   
¬A B ¬A B
We can now close some branches due to contradictions:
√ [A → B]
√ [A v D]
√ [C ^ ¬B]

________
 
A D
C C
¬B ¬B
______ ______
   
¬A B ¬A B
When we close a branch it tells us that the branch in question does not represent a way in which all the original sentences could be true. We find that there is only one possible way in which the original sentences could be consistent  that is if "D", "C", "B", and "A" were true. But we cannot say that the sentences are consistent. Let's look at what the tableau says. It says that the sentences could be true if "John's father is Scottish", "John's father is English", "John doesn't like tea", and "John is not English". We must decide here if it is possible to be both English and Scottish. Strictly, it is not, so we must conclude that the sentences are inconsistent.
There is a better way of testing validity  constructing a tableau using a "counterexample set". A counterexample set is produced simply by negating the conclusion of an argument. In logical analysis, we usually test arguments for validity instead of simple sets of statements and testing the counterexample set of an argument allows us to do that.
Lets take the following argument, already formalised in the propositional calculus (it does not matter for our purposes what the symbols stand for):
[P ↔ [¬Q ^ R]], [P → [Q v S]]. Therefore, [[Q ^ R] → S]
By
negating the conclusion ("[[Q ^ R] → S]") we can produce the counterexample set:
[P ↔ [Q ^ R]]
[P → [¬Q v S]]
¬[[Q ^ R] → S]
We can then construct the following tableau. This is just one example, there are several ways to do it:
[P ↔ [Q ^ R]]
[P → [¬Q v S]]
¬[[Q ^ R] → S]
[Q ^ R]
¬S
____________________
 
P ¬P
[Q ^ R] ¬[Q ^ R]
Q
R
__________________
 
¬P [¬Q v R]
____________
 
¬Q R
The tableau closes, so the counterexample set is inconsistent. If the counterexample set is inconsistent, then the original argument must be consistent, and therefore, it must be a valid argument.
Wilfred Hodges, "Logic"
http://logic.philosophy.ox.ac.uk